---
title: Pseudotrace Construction in Algebra & VOAs
url: https://www.emergentmind.com/topics/pseudotrace-construction
type: topic
---

# Pseudotrace Construction in Algebra & VOAs

Pseudotrace construction is a trace-like procedure that starts from a symmetric linear functional on an algebra and produces, via a projective module or projective generator, a symmetric linear functional on an endomorphism algebra or on a related algebra of operators. In the finite-dimensional setting, it is the composition of the Hattori–Stallings universal trace with a symmetric linear functional; in the almost unital and finite-dimensional setting, it is formulated using local units and coordinate systems; and in the vertex-operator-algebra setting, it is identified with vacuum torus conformal blocks through an end/coend object and sewing-factorization. The recent literature shows that these formulations are not separate constructions but compatible realizations of the same basic mechanism [1001.2696], [2508.00431], [2508.04532].

## 1. Classical algebraic definition

In the finite-dimensional associative-algebra setting, let $A$ be a finite-dimensional associative algebra over an algebraically closed field of characteristic $0$, and let $\phi \in \mathrm{SLF}(A)$ be a symmetric linear function, meaning
\[
\phi(ab)=\phi(ba)\qquad \text{for all } a,b\in A.
\]
If $P_A$ is a finitely generated projective right $A$-module, then $P$ admits an $A$-coordinate system $\{u_i,\alpha_i\}$ with $u_i\in P$ and $\alpha_i\in \operatorname{Hom}_A(P,A)$ such that
\[
\sum_i u_i\,\alpha_i=\operatorname{id}_P.
\]
The Hattori–Stallings trace is then
\[
T_P:\operatorname{End}_A(P)\longrightarrow A/[A,A],\qquad 
T_P(f):=\Big[\sum_i \alpha_i\big(f(u_i)\big)\Big].
\]
Composing with $\phi$ gives the induced symmetric linear function
\[
\phi_P:=\phi\circ T_P,\qquad 
\phi_P(f)=\phi\!\left(\sum_i \alpha_i\big(f(u_i)\big)\right),
\]
and $\phi_P(fg)=\phi_P(gf)$ for all $f,g\in \operatorname{End}_A(P)$ [1001.2696].

This formulation makes precise the sense in which a pseudotrace is a trace only after passage through a symmetric linear functional on the ground algebra. The universal part is $T_P$, which lands in $A/[A,A]=HH_0(A)$; the scalar-valued part is the evaluation against $\phi$. The construction is independent of the chosen coordinate system because differences between coordinate-system expressions lie in $[A,A]$, which is annihilated by every symmetric linear functional [1001.2696].

In the literature summarized here, this is the basic algebraic template from which later generalizations proceed. A central point is that pseudotraces are not arbitrary linear forms on endomorphism rings: they are induced functorially from cyclic classes in the source algebra and therefore inherit cyclicity, additivity on direct sums, and compatibility with Morita-type passage to basic algebras [1001.2696].

## 2. Coordinate systems, projectivity, and Miyamoto’s formulation

The construction becomes especially explicit when the algebra is symmetric. In the setting of a basic indecomposable symmetric algebra $P$ with primitive idempotents $e_1,\dots,e_k$ and a symmetric linear function $\phi\in \mathrm{SLF}(P)$ inducing a nondegenerate associative symmetric bilinear form, Arike shows that Miyamoto’s pseudotrace map is exactly the induced symmetric linear function $\phi_W$ on $\operatorname{End}_P(W)$ for finitely generated projective modules $W$ [1001.2696].

The key structural notion in that setting is that of an interlocked module. A finitely generated right $P$-module $W_P$ is interlocked with $\phi$ if, for each $i$,
\[
\ker(f_i)=\bigoplus_{\rho\in \Omega\setminus\{e_i\}} W\rho,
\]
where $\Omega$ is the canonical basis constructed in the paper and $f_i$ are the socle basis elements dual to the primitive idempotents. The main result is that $W$ is interlocked with $\phi$ if and only if $W$ is projective, and the multiplicity of the indecomposable projective $e_iP$ in $W$ is $\dim_k Wf_i$ [1001.2696].

For projective $W_P\simeq \bigoplus_{i=1}^k (e_iP)^{\oplus n_i}$, one chooses basis elements $v_{i,t}$ corresponding to the summand generators and defines maps $\alpha_{i,t}\in \operatorname{Hom}_P(W,P)$. This yields a $P$-coordinate system, and Miyamoto’s pseudotrace is
\[
\operatorname{tr}^{\mathrm{Miy}}_\phi(a)
=
\sum_{i=1}^k \sum_{t=1}^{n_i}
\phi\!\big(\alpha_{i,t}(a(v_{i,t}))\big).
\]
Arike proves
\[
\phi_W=\operatorname{tr}^{\mathrm{Miy}}_\phi
\quad \text{on } \operatorname{End}_P(W),
\]
thereby identifying the pseudotrace map with the classical coordinate-system construction [1001.2696].

This equivalence dispels a common misconception that Miyamoto’s pseudotraces are intrinsically tied to a special basis-level combinatorial formula. The basis formula is present, but the paper shows that it is a special case of the Hattori–Stallings/symmetric-linear-function mechanism. A plausible implication is that many formal properties of pseudotraces are better understood at the level of $A/[A,A]$ and Morita invariance than at the level of basis expansions alone.

## 3. Generalization to almost unital and finite-dimensional algebras

The 2025 generalization by Gui–Zhang extends pseudotraces from unital finite-dimensional algebras to almost unital and finite-dimensional (AUF) algebras, which may be non-unital or infinite-dimensional as vector spaces but possess sufficiently many idempotents [2508.00431]. An associative algebra $A$ is almost unital if every element has a local idempotent unit and finite sets of idempotents admit a common dominating idempotent. It is AUF if there exists a family of mutually orthogonal idempotents $(e_i)_{i\in I}$ such that
\[
\dim_{\mathbb C}(e_iAe_j)<\infty
\quad\text{for all } i,j\in I,
\]
and
\[
A=\bigoplus_{i,j\in I} e_iAe_j.
\]

The natural finiteness category is $\mathrm{Coh}(A)$, the category of coherent left $A$-modules: finitely generated modules that are quotients of finite direct sums of modules of the form $Ae_i$. If $A$ is strongly AUF and $G\in \mathrm{Coh}(A)$ is a projective generator, then
\[
B:=\operatorname{End}_{A,-}(G)^{\mathrm{opp}}
\]
is the endomorphism algebra controlling the right action on $G$ [2508.00431].

The left-coordinate-system version of pseudotrace is formulated for an $A$–$B$ bimodule $M$ that is projective as a right $B$-module. A left coordinate system is a family
\[
\alpha_i\in \operatorname{Hom}_{-,B}(B,M),\qquad
\alpha^i\in \operatorname{Hom}_{-,B}(M,B),
\]
satisfying local finiteness and
\[
\sum_{i\in I}\alpha_i\circ \alpha^i=\operatorname{id}_M.
\]
Then the left $B$-trace is
\[
\operatorname{Tr}^B(x):=\sum_{i\in I}\alpha^i\circ x\circ \alpha_i
\in B/[B,B],
\]
and for $\phi\in \mathrm{SLF}(B)$ the pseudotrace is
\[
\operatorname{Tr}^\phi(x)=
\sum_{i\in I}\phi\big(\alpha^i\circ x\circ \alpha_i(1_B)\big).
\]
It is independent of the chosen coordinate system, and $\operatorname{Tr}^\phi\in \mathrm{SLF}(A)$ [2508.00431].

The theory also includes a right-coordinate-system construction yielding the map in the opposite direction,
\[
\psi\in \mathrm{SLF}(A)\longmapsto {}^{\psi}\operatorname{Tr}\in \mathrm{SLF}(B),
\]
and under the hypotheses that $A$ is strongly AUF and $G$ is a projective generator, these two constructions are inverse linear isomorphisms
\[
\mathrm{SLF}(A)\simeq \mathrm{SLF}(B).
\]
The non-degeneracies on the two sides are equivalent as well [2508.00431].

This result is a Morita-type invariance statement for spaces of symmetric linear functionals, but formulated in the non-unital AUF context rather than the ordinary finite-dimensional unital context. Because the algebra may be infinite-dimensional while still decomposing into finite-dimensional blocks, the construction isolates the genuinely finite part needed for trace theory: local units, coherent modules, and blockwise finite support.

## 4. Ends, coends, and the universal algebra in VOA theory

In the vertex-operator-algebra setting, the construction is recast geometrically and categorically. Let
\[
\mathbb V=\bigoplus_{n\in \mathbb N}\mathbb V(n)
\]
be an $\mathbb N$-graded $C_2$-cofinite VOA, not necessarily rational or self-dual. Then $\mathrm{Mod}(\mathbb V)$, the linear category of grading-restricted generalized $\mathbb V$-modules, is finite abelian. For a grading-restricted generalized $\mathbb V^{\otimes N}$-module $\mathcal M$, the contragredient module is
\[
\mathcal M' := \bigoplus_{\lambda_\bullet\in\mathbb C^N} (\mathcal M_{[\lambda_\bullet]})^*,
\]
and the paper studies the end
\[
\mathbb E:=\int_{\mathbb M\in \mathrm{Mod}(\mathbb V)} \mathbb M\otimes_{\mathbb C}\mathbb M'
\in \mathrm{Mod}(\mathbb V^{\otimes 2}).
\]
The main identification is
\[
\mathbb E \simeq \boxtimes_N,
\]
where $\boxtimes_N$ is the default fusion product of $\mathbb V$ along the standard two-pointed sphere $N$, and dually
\[
{}_N \simeq \int^{\mathbb M\in \mathrm{Mod}(\mathbb V)} \mathbb M' \otimes \mathbb M
\]
as a coend [2508.04532].

For each module $\mathbb M$ there is a canonical $\mathbb V^{\otimes 2}$-morphism
\[
\pi_{\mathbb M}:\boxtimes_N\to \operatorname{End}^0(\mathbb M)\simeq \mathbb M\otimes \mathbb M',
\]
and the family $(\pi_{\mathbb M})_{\mathbb M}$ is dinatural and universal. This realizes $\mathbb E$ as the universal pairing object for modules and their contragredients [2508.04532].

The paper then equips $\boxtimes_N$, and therefore $\mathbb E$, with a natural associative $\mathbb C$-algebra structure compatible with the $\mathbb V^{\otimes 2}$-module structure. Using the canonical conformal blocks
\[
\Phi=\Phi_{+,+}:\boxtimes_N\otimes X\to X,
\qquad
\Psi=\Phi_{-,-}:X\otimes \boxtimes_N\to X,
\]
the multiplication is
\[
\psi_1\diamond \psi_2:=\Phi(\psi_1\otimes \psi_2)=\Psi(\psi_1\otimes \psi_2).
\]
Associativity follows from a special case of the sewing-factorization theorem, which equates different parenthesizations of conformal-block compositions [2508.04532].

There is also a canonical involution
\[
\Theta:\boxtimes_N\to \boxtimes_N,\qquad \Theta^2=\operatorname{id},
\]
satisfying
\[
\Theta Y_+(v)_n = Y_-(v)_n\Theta,\qquad
\Theta Y_-(v)_n = Y_+(v)_n\Theta,
\]
and $\Theta$ is an anti-automorphism of the algebra:
\[
\Theta\psi_1\diamond \Theta\psi_2=\Theta(\psi_2\diamond \psi_1).
\]
With the idempotents
\[
\chi_\lambda=P_+(\lambda)P_-(\lambda)\alpha_1^\sharp(1),
\]
the algebra decomposes into finite-dimensional corners:
\[
\chi_\lambda\diamond \chi_\mu=\delta_{\lambda,\mu}\chi_\mu,
\qquad
\dim(\chi_\lambda\diamond \boxtimes_N\diamond \chi_\mu)<\infty,
\]
and every $\psi\in \boxtimes_N$ is a finite sum
\[
\psi=\sum_{\lambda,\mu}\chi_\lambda\diamond \psi \diamond \chi_\mu.
\]
Thus $\boxtimes_N$ is an almost-unital finite-dimensional algebra in the sense of Gui–Zhang [2508.04532].

## 5. Symmetric linear functionals, pseudo-\(q\)-traces, and torus blocks

The bridge from the universal algebra $\mathbb E$ to conformal blocks is given by symmetric linear functionals. For a $\mathbb V^{\otimes 2}$-module $X$,
\[
\mathrm{SLF}(X):=
\{\varphi\in X^*\mid 
\varphi(\psi\diamond_L w)=\varphi(w\diamond_R \psi)\ \forall\psi\in \boxtimes_N,\ w\in X\}.
\]
The paper proves that
\[
\mathrm{SLF}(X)=T_C^*(X),
\]
that is, symmetric linear functionals on $X$ are precisely conformal blocks on the standard sphere $C$. Specializing to $X=\boxtimes_N$ gives
\[
\mathrm{SLF}(\boxtimes_N)=T_C^*(\boxtimes_N),
\]
and sewing with the canonical insertion
\[
\alpha_{z,q}^\sharp=q^{L_+(0)}\circ \alpha_z^\sharp:\mathbb V\to \mathbb E
\]
yields the isomorphism
\[
\mathrm{SLF}(\mathbb E)\simeq T^*_{T_{z,q}}(\mathbb V),
\qquad
\xi\mapsto \xi\circ \alpha_{z,q}^\sharp,
\]
where the target is the space of vacuum torus conformal blocks [2508.04532].

Now let $\mathbb G$ be a projective generator in $\mathrm{Mod}(\mathbb V)$ and set
\[
B=\operatorname{End}_{\mathbb V}(\mathbb G)^{\mathrm{opp}}.
\]
Because $\mathbb G$ is a projective right $B$-module, one may choose a left coordinate system
\[
(\alpha_i,\alpha^i)_{i\in I}
\]
with
\[
\sum_i \alpha_i\alpha^i(\xi)=\xi
\]
and the required finiteness properties. For $\phi\in \mathrm{SLF}(B)$, the pseudotrace on $\operatorname{End}^0_B(\mathbb G)$ is
\[
\operatorname{Tr}^\phi(x)=
\sum_{i\in I}\phi\big(\alpha^i\circ x\circ \alpha_i(1_B)\big).
\]
This is independent of the choice of left coordinate system, and the main theorem of Gui–Zhang gives a linear isomorphism
\[
\mathrm{SLF}(B)\simeq \mathrm{SLF}(\mathbb E),
\qquad
\phi\mapsto \operatorname{Tr}^\phi\circ \pi_{\mathbb G}
\]
preserving non-degeneracy [2508.00431], [2508.04532].

Composing with the sewing-factorization isomorphism gives the pseudo-$q$-trace construction
\[
\Phi_{\mathbb G}:
\mathrm{SLF}(\operatorname{End}_{\mathbb V}(\mathbb G)^{\mathrm{opp}})
\longrightarrow
\mathrm{VTCB}(\mathbb V),
\]
and explicitly
\[
\Phi_{\mathbb G}(\phi)(v)=
\sum_{\lambda\in \mathbb C}
\operatorname{Tr}^\phi\!\big(P(\lambda)Y_{\mathbb G}(v,z)q^{L(0)}P(\lambda)\big).
\]
Equivalently,
\[
\operatorname{ptr}^{(q)}_{\mathbb G,\phi}(v;\tau)
:=
\sum_{\lambda\in\mathbb C}
\operatorname{Tr}^\phi\!\big(P(\lambda)Y_{\mathbb G}(v,z)q^{L(0)}P(\lambda)\big),
\]
where $q=e^{2\pi i\tau}$ and the sum converges absolutely by $C_2$-cofiniteness and grading restrictions [2508.04532].

The paper uses the normalization without an explicit $-c/24$ shift. It states that incorporating $q^{L(0)-c/24}$ is a standard physics normalization, whereas the geometric sewing choice here implements $q^{L(0)}$ via the outgoing local coordinate $1/(q\zeta)$ [2508.04532].

## 6. The main isomorphism, conjectures, and scope of the term

The central VOA theorem proves the Gainutdinov–Runkel conjecture: for any projective generator $\mathbb G$ in $\mathrm{Mod}(\mathbb V)$, the pseudo-$q$-trace construction yields a linear isomorphism
\[
\Phi_{\mathbb G}:
\mathrm{SLF}(\operatorname{End}_{\mathbb V}(\mathbb G)^{\mathrm{opp}})
\longrightarrow
\mathrm{VTCB}(\mathbb V).
\]
The proof combines three ingredients: the AUF pseudotrace isomorphism $\mathrm{SLF}(B)\simeq \mathrm{SLF}(\mathbb E)$, the identification $\mathrm{SLF}(\mathbb E)=T_C^*(\mathbb E)$, and the sewing-factorization isomorphism from sphere blocks to torus blocks [2508.04532].

A corollary confirms the Arike–Nagatomo conjecture. If $A$ is a unital finite-dimensional $\mathbb C$-algebra such that $\mathrm{Mod}(A)\simeq \mathrm{Mod}(\mathbb V)$ as linear categories, then
\[
\mathrm{SLF}(A)\simeq \mathrm{VTCB}(\mathbb V).
\]
The proof uses the end/coend description together with the identification of the Deligne tensor product with finite-dimensional bimodules and the realization
\[
\operatorname{Hom}_{\mathrm{Bim}^f(A)}(A,A^*)\simeq \mathrm{SLF}(A)
\]
cited in the paper [2508.04532].

The same framework yields geometric interpretations of other algebraic structures. The Zhu algebra $A(\mathbb V)$ and the higher Zhu algebras $A_n(\mathbb V)$ can be realized as quotients of $\mathbb E$, with their multiplications encoded by the $\mathbb V^{\otimes 2}$-module structure of $\mathbb E$ and truncations by the idempotents $\chi_\lambda$ [2508.04532]. In strongly-finite logarithmic CFT, when $\mathbb V$ is self-dual, $\mathbb V(0)$ is finite-dimensional, and $\mathrm{Mod}(\mathbb V)$ is rigid and factorizable, a distinguished modified trace exists, unique up to scale and non-degenerate; it yields an isomorphism
\[
T^\phi:\mathbb E\simeq {}_N,
\]
which the paper states is consistent with unimodularity [2508.04532].

The literature also shows that the term “pseudotrace” is not entirely uniform across fields. In cosmology, a 2020 paper defines a “pseudotrace” density by
\[
\bar\theta(T)\equiv e(T)-\frac{1}{c_{s,b}^2(T)}\,p(T),
\]
with phase difference
\[
\Delta\bar\theta(T)=\Delta e(T)-\frac{1}{c_{s,b}^2(T)}\,\Delta p(T),
\]
to describe the hydrodynamics of first-order phase transitions [2004.06995]. That usage is formally unrelated to the algebraic pseudotrace construction discussed above. This suggests that, in current arXiv usage, “pseudotrace” may denote either a trace-like algebraic functional built from symmetric linear forms and projective modules, or a specific thermodynamic linear combination of energy density and pressure, and the distinction is terminologically important.

Across the algebraic and VOA settings, however, the construction has a consistent conceptual core: a universal cyclic functional is transported through a projective generator and then identified either with a symmetric linear functional space or, after sewing, with a space of conformal blocks. In the formulation of the 2025 VOA paper, this is summarized by the chain
\[
\mathrm{SLF}(\operatorname{End}_{\mathbb V}(\mathbb G)^{\mathrm{opp}})
\;\simeq\;
\mathrm{SLF}(\mathbb E)
\;\simeq\;
\mathrm{VTCB}(\mathbb V),
\]
which geometrically explains why pseudo-$q$-traces give all vacuum torus conformal blocks for $C_2$-cofinite VOAs without assuming rationality or self-duality [2508.04532].

Source: https://www.emergentmind.com/topics/pseudotrace-construction