Symmetric Operator Algebras

• Symmetric operator algebras are nonselfadjoint operator algebras that are completely isometrically isomorphic to their transpose, generalizing commutative C*-algebras.
• They exhibit 3-commutativity, meaning that the product of any three elements is independent of the order, a property intrinsically linked to their reversible structure.
• Injective envelope methods and explicit examples, such as those from canonical anticommutation relations, are used to investigate conditions for full commutativity and structural classification.

A symmetric operator algebra is a nonselfadjoint operator algebra whose matrix norm structure is completely isometrically isomorphic to that of its opposite or “transpose” algebra, i.e., the subalgebra of $B(H)$ on which the entrywise transpose is a complete isometry. These algebras generalize the notion of commutative $C^*$-algebras and are intimately connected with reversible operator algebras: operator algebras that, when reversed in multiplication, again yield an abstract operator algebra in the operator space sense. Understanding the spectrum from full commutativity to the higher-order “3-commutativity” property is central to their structure theory and applications, especially in the nonunital setting (Blecher, 22 Sep 2025).

1. Reversible and Symmetric Operator Algebras: Definitions and Interrelations

A reversible operator algebra is one that remains an operator algebra under reversed multiplication; more formally, $(A, \circ)$, where $x \circ y = yx$, has an operator space structure for which the norm of $[a_{ij}]$ equals that of $[a_{ji}]$. A symmetric operator algebra is a subalgebra of $B(H)$ such that, for all $[a_{ij}]$, the norm of $[a_{ij}]$ equals that of its transpose $[a_{ji}]$. These classes coincide with the completely isometric identification of an operator algebra with its opposite via the transposition (see Theorem 4.2 and related discussions).

This fundamental symmetry gives rise to a number of equivalent conditions in many cases: commutativity, symmetry, reversibility, the operator space structure of the opposite, and existence of completely isometric representations by symmetric matrices. The relationships become particularly intricate in the absence of a unit, where subtleties concerning injective envelopes, triple product representations, and corners of $C^*$-algebras emerge.

2. Main Structural Results: 3-Commutativity and Commutativity Criteria

A principal finding is that every reversible (and hence every symmetric) operator algebra is at least 3-commutative—for any three or more elements, the order of multiplication is irrelevant. Specifically, if $A$ is reversible, then for all $a_1, a_2, a_3 \in A$,

$$a_1(a_2 a_3) = (a_1 a_2) a_3 = (a_2 a_1) a_3 = a_1(a_3 a_2) = \cdots$$

This property extends to all products of three or more elements, regardless of the absence of full commutativity. This 3-commutativity is proved using technical analysis involving the injective envelope: any $A$ can be embedded into an injective operator system $I(A)$ (a sub-TRO of $B(K)$), and for $A$ reversible, there exist distinguished elements $z, w \in I(A)$ so that $xy = xz^*y$ and $yx = xw^*y$ for all $x, y \in A$. The commutator ideal and related “shadow algebras” arising from these dual multipliers become commutative subalgebras (Blecher, 22 Sep 2025).

Full commutativity arises if and only if $z = w$, that is, left and right multiplication structures coincide in $I(A)$. Several sufficient conditions force commutativity: idempotence (the square is dense), faithful module actions (i.e., “c-faithfulness”), or being “essential” in its injective envelope (i.e., the unitization $A^1$ forms an essential extension).

3. Injective Envelope Techniques and Technical Machinery

The use of the injective envelope $I(A)$ is pivotal in analyzing the structure of symmetric operator algebras, especially nonunital ones. $I(A)$ is a unique (up to complete isometry) injective object containing $A$, often realized as a TRO (ternary ring of operators). Via a standard position (a specific representation in $B(K)$), the product in $A$ is expressed as $xy = xz^*y$ for a unique contraction $z$ in $I(A)$. Reversibility then forces the existence of another element $w$ such that $yx = xw^*y$ for all $x, y$.

When $z = w$, $A$ is commutative. If not, $A$ is always 3-commutative. This machinery is particularly useful for explicitly computing in low-dimensional matrix settings and for elucidating why symmetric properties do not always yield full commutativity.

4. Examples, Counterexamples, and Canonical Anticommutation Relations

Several concrete examples are constructed using the canonical anticommutation relations (CAR), a structure fundamental in mathematical physics. Algebras generated by two partial isometries $S, T$ with $ST + TS = 0$ can be realized as reversible (i.e., symmetric) but not commutative operator algebras, indicating that 3-commutativity is genuinely weaker than commutativity. In such examples, the dual elements $z \neq w$ in $I(A)$ can be explicitly identified (e.g., as projections in $M_4$ for appropriate $A \subset M_4$). The transpose map or the automorphism $x \mapsto -x^T$ provides the explicit isometric involution in these settings.

These counterexamples highlight that even in dimension 3 or 4, the distinction between reversible/symmetric and commutative algebras is profound, and further motivate the paper of the commutator ideal and triangularizability (in the sense of Radjavi–Rosenthal) for such operator algebras.

5. Sufficient Conditions for Commutativity and Open Problems

A comprehensive table of equivalences is established for reversible algebras where idempotence, faithfulness, or essentiality in the injective envelope hold:

Property	Implies Commutativity	Key references
Idempotence	Yes	Theorem mainc
Left/right faithfulness	Yes	Theorem mainc
Essential extension	Yes	Theorem 4.4
3-commutativity alone	Not always	Section 5 Examples

However, without these properties (notably in the nonunital case), 3-commutativity can exist without full commutativity, as shown by explicit CAR examples.

Open questions identified include: the characterization of all (finite-dimensional) reversible subalgebras of $M_n$ ($n = 3, 4$), the status of rigidity vs. essentiality in extensions, and low-dimensional complete classifications. There is also interest in the algebraic structure of the commutator ideal and its action, with connections to Lie theory and triangularizability (Blecher, 22 Sep 2025).

6. Implications and Research Directions

The identification of 3-commutativity as a general property of symmetric operator algebras provides a new tool for structural analysis, especially in nonselfadjoint, nonunital, and noncommutative settings. The injective envelope approach reveals deep connections with operator system theory, TROs, and Morita equivalence phenomena. The construction of explicit counterexamples using CAR further shapes the boundary between symmetry and commutativity, and motivates refined invariants (such as the pairs $(z, w)$ in $I(A)$).

Potential applications arise in classification problems in operator algebra theory, quantum information (where symmetry under transposition and tensor products is critical), and the representation theory of operator algebras arising in physics. The outstanding computational problems in low-dimensions suggest further investigations accessible both theoretically and computationally.

7. Summary Table: Symmetry, Reversibility, and Commutativity

Notion	Definition	When Implies Commutativity
Symmetric	Transpose is a complete isometry	Idempotent/faithful/essential case
Reversible	Algebra with reversed multiplication is an operator algebra	Idempotent/faithful/essential case
3-Commutative	Products of three or more elements commute as for a commutative algebra	General
Fully Commutative	$xy = yx$ for all $x, y$	If $z = w$ in $I(A)$

In conclusion, symmetric operator algebras form a strictly larger class than commutative ones, distinguished by their 3-commutativity and deep structural ties to the injective envelope and anti-commutation phenomena. Their paper reveals rich operator-algebraic, topological, and representation-theoretic properties, many of which remain open for further exploration (Blecher, 22 Sep 2025).