---
title: Similarity to Isometric Semigroups
url: https://www.emergentmind.com/topics/similarity-to-isometric-semigroups
type: topic
---

# Similarity to Isometric Semigroups

Similarity to isometric semigroups concerns the passage from a given semigroup, or a representation of a semigroup, to one implemented by isometries after an appropriate change of coordinates, typically by an invertible operator. In the literature considered here, the theme appears in several technically distinct but related forms: simultaneous similarity of matrix semigroups to semigroups of partial isometries [1306.1974], similarity of Hilbert-space $C_0$-semigroups to isometric semigroups via uniform two-sided norm bounds [2509.01007], Følner-type criteria for amenable semigroups [1804.08960], dilation and Wold-type decompositions for semigroup representations [1710.09004], [2103.02070], and $C^*$-algebraic realizations in which partial-isometric crossed products occur as full corners of larger crossed products and hence are Morita equivalent to group crossed products [1710.06708]. Across these settings, the decisive issues are the control of norms, the behavior of idempotents and projections, the availability of inverse or Nica-covariant structure, and the extent to which semigroup data can be embedded into group-like or unitary models.

## 1. Operator-theoretic formulations and analytic criteria

For Hilbert space $C_0$-semigroups, the basic criterion is the classical characterization attributed in the survey material to Sz.-Nagy: a $C_0$-semigroup $\mathcal{T}=(T(t))_{t\ge 0}$ is similar to a semigroup of isometries if and only if there exist constants $\alpha,\beta>0$ such that
$$
\alpha \|h\| \leq \|T(t)h\| \leq \beta \|h\|, \qquad \forall h\in H,\ \forall t\ge 0.
$$
If $\mathcal{T}$ is invertible, this is equivalent to similarity to a unitary group [2509.01007]. The same source emphasizes a structural asymmetry between contraction similarity and isometric similarity: for contractions, an upper bound suffices, whereas for isometries both lower and upper bounds are necessary.

The analytic criteria extend beyond pointwise norm estimates. One formulation uses weighted orbit means: for a generator $E$, similarity to an isometric $C_0$-semigroup is equivalent to the existence of a Hilbert space $K$ and a densely defined operator $C:H\to K$ such that, for some $\alpha,\beta>0$,
$$
\alpha \|h\|^2 \leq \limsup_{t\to\infty}\frac{1}{t}\int_0^t \|CT(s)h\|_K^2\,ds \leq \beta \|h\|^2,
$$
or the same with $\liminf$ [2509.01007]. A resolvent counterpart replaces time averages by Abel-type integrals of $(\varepsilon+i\xi-E)^{-1}$, again with two-sided bounds. These criteria formalize the idea that similarity to an isometric semigroup is encoded by norm equivalence visible either in long-time averages or in resolvent growth.

For discrete dynamics and amenable semigroups, Cesàro and Følner averages play the same role. A single operator $T$ is similar to an isometry if there exist $m,M>0$ such that
$$
m^2\|x\|^2 \leq \frac{1}{N}\sum_{n=0}^{N-1}\|T^n x\|^2 \leq M^2\|x\|^2
$$
for every $N\ge 1$ and every $x\in H$ [1804.08960]. For a countable discrete semigroup satisfying the Strong Følner Condition, analogous lower and upper bounds over a right Følner sequence, together with asymptotic invariance over symmetric differences $F_Ns\triangle F_N$, imply similarity to an isometric representation [1804.08960]. In this sense, amenability supplies averaging devices from which an equivalent Hilbertian norm can be reconstructed.

## 2. Matrix semigroups, partial isometries, and inverse structure

In finite dimensions, the decisive result is Popov’s characterization of irreducible norm closed semigroups of complex matrices. If $\mathcal{S}$ is an irreducible norm closed semigroup of complex $n\times n$ matrices, then the following are equivalent: $\mathcal{S}$ is simultaneously similar to a semigroup of partial isometries; all idempotents in $\mathcal{S}$ commute and every spectrum satisfies $\sigma(T)\subseteq \{0\}\cup\mathbb{T}$; and all idempotents in $\mathcal{S}$ commute while there exist constants $c_1,c_2>0$ such that
$$
c_1 \leq \|T\| \leq c_2
$$
for every nonzero $T\in\mathcal{S}$ [1306.1974]. This theorem generalizes the classical bounded-group criterion for similarity to unitaries.

Two features of this characterization are especially significant. First, the algebraic condition on idempotents is indispensable: the commuting of projections is a structural remnant of partial-isometric behavior. Second, norm closedness is essential. The counterexample described in the source material exhibits an irreducible semigroup with spectrum in $\{0\}\cup\mathbb{T}$ and norm $1$ on nonzero elements whose norm closure contains non-commuting idempotents, so the semigroup is not similar to a semigroup of partial isometries [1306.1974]. A common misconception is therefore that spectral control alone should suffice; the theorem shows that the lattice-theoretic behavior of idempotents is equally fundamental.

A broader structural theorem identifies self-adjoint semigroups of partial isometries with faithful representations of abstract inverse semigroups [1306.1973]. In the irreducible case, every such semigroup acts as a family of generalized weighted composition operators on a space $L^2(\Omega,K)$. If the semigroup contains a compact operator, then the measure space is purely atomic and the semigroup is represented by “zero-unitary” matrices, meaning block matrices with at most one nonzero unitary block in each row and column [1306.1973]. This places groups of unitaries as a special case inside a much larger inverse-semigroup framework.

## 3. Ordered semigroups and rigidity of isometric representations

For subsemigroups of abelian torsion-free groups, total order governs the rigidity of isometric representations. If $S\subseteq G$, the following are equivalent: $S$ is a positive cone; all $C^*$-algebras generated by faithful isometric non-unitary representations of $S$ are canonically isomorphic; all such representations are inverse; and, for every such representation $V$ and all relevant semigroup elements,
$$
V_aV_b^*V_cV_d^* = V_b^*V_aV_cV_d^*
$$
[1203.5490]. The same source states the corollary that the universal semigroup $C^*$-algebra coincides with the reduced semigroup $C^*$-algebra if and only if $S$ is totally ordered.

The case $G=\mathbb{Z}\times\mathbb{Z}$ shows that even within total orders there is a sharp dichotomy. If the positive cone induces a total archimedean order, then there exist at least two unitarily inequivalent irreducible faithful isometric representations. If the order is lexicographical-product order, then all such representations are unitarily equivalent [1203.5490]. Thus total order is necessary for canonical isomorphism of all representation-generated algebras, but it does not by itself enforce uniqueness of irreducible representations.

The perforated semigroup $\mathbb{Z}_+\setminus\{1\}$ provides a complementary example. Its non-unitary irreducible isometric inverse representations are classified up to unitary equivalence by two models, $T_0$ and $T_1$, and every isometric inverse representation decomposes as
$$
\pi = kT_0 \oplus lT_1 \oplus \pi_2,
$$
where $\pi_2$ is unitary [1303.0387]. At the same time, there is a continuum of non-inverse irreducible isometric $\beta$-representations, with $T_\beta$ inverse if and only if $\beta=0$ or $|\beta|=1$ [1303.0387]. These examples show that once total order or inverse structure is lost, the landscape of isometric representations becomes much less rigid.

## 4. Dilations, decompositions, and obstructions

One route to similarity is through dilation theory. For a unital representation $T$ of a right LCM semigroup, three conditions are equivalent: $T$ has a $*$-regular dilation; $T$ has a minimal isometric Nica-covariant dilation; and, for every finite subset $F$,
$$
Z(F)=\sum_{U\subseteq F}(-1)^{|U|}TT^*(\vee U)\ge 0.
$$
This Brehmer-type condition gives an explicit positivity test for passage to an isometric Nica-covariant model [1710.09004]. The result generalizes regular dilation theorems from more restrictive semigroup classes and identifies a concrete interface between semigroup combinatorics and operator positivity.

A complementary viewpoint is provided by Wold-type decompositions. For an isometric representation $\{W,V_1,\dots,V_n\}$ of the odometer semigroup, there is a unique decomposition
$$
H = H_{uu}\oplus H_{us}\oplus H_{su}\oplus H_{ws},
$$
where the four summands encode unitary-row unitary, unitary-pure, pure-row unitary, and weak bi-shift behavior [2103.02070]. In the isometric Nica-covariant case, the weak bi-shift component is replaced by a summand unitarily equivalent to direct sums of the left regular representation, yielding
$$
H = H_{uu}\oplus H_{us}\oplus H_{su}\oplus H_{ss}
$$
[2103.02070]. The same source states that similarity to a direct sum of unitary and pure representations occurs exactly when the weak bi-shift, or left-regular, component is trivial.

Not every generalized isometric notion leads back to true isometric similarity. For $m$-isometric $C_0$-semigroups, the generator is characterized by a Lumer–Phillips type theorem involving closedness, surjectivity, quasidissipativity, and $m$-skew-symmetry [1809.00664]. However, for $m>1$, these semigroups are not generally similar to isometric ones. The prototypical analytic $2$-isometry $M_z$ on the Dirichlet space is not similar to any isometry on a Hilbert space because its spectrum is the closed unit disk [1809.00664]. A standard misunderstanding is therefore that higher-order isometricity should behave like genuine isometricity; the Dirichlet-space model shows that it does not.

## 5. Crossed products, full corners, and Morita-equivalent group models

In the $C^*$-algebraic setting, similarity to isometric semigroups is often replaced by a corner or Morita-equivalence construction. Let $\Gamma^+$ be the positive cone of a totally ordered abelian discrete group $\Gamma$, and let $\alpha$ be an action of $\Gamma^+$ by extendible endomorphisms of a $C^*$-algebra $A$. The main theorem of the 2017 crossed-product paper states that the partial-isometric crossed product satisfies
$$
A\times_{\alpha}^{\mathrm{piso}\Gamma^+} \cong q(\mathcal{B}\times_\beta \Gamma)q,
$$
where $\mathcal{B}$ is a subalgebra of $\ell^\infty(\Gamma,A)$ generated by faithful copies of $A$, $\beta$ is induced by shift on $\ell^\infty(\Gamma,A)$, and $q$ is a projection in the multiplier algebra [1710.06708]. Consequently, the partial-isometric crossed product is Morita equivalent to a group crossed product. The same paper identifies an essential ideal $J$ in $A\times_{\alpha}^{\mathrm{piso}\Gamma^+}$, namely the kernel of the surjection onto the isometric crossed product, and shows that $J$ is a full corner in an ideal $\mathcal{I}\times_\beta\Gamma$ of $\mathcal{B}\times_\beta\Gamma$ [1710.06708].

An earlier full-corner theorem provides the semigroup-side model more directly. For the same kind of system, the partial-isometric crossed product is isomorphic to a full corner in a subalgebra of $\mathcal{L}(\ell^2(\Gamma^+,A))$, and there is always a natural surjection
$$
A\times_{\alpha}^{\mathrm{piso}\Gamma^+} \to A\times_{\alpha}^{\mathrm{iso}\Gamma^+}.
$$
Its kernel is explicitly described, and for $\Gamma^+=\mathbb{N}$ or $\mathbb{R}_+$ it is a full corner of the compact operators on the Hilbert module [1309.2363]. In the automorphism case, the partial-isometric crossed product becomes an isometric crossed product and hence a full corner in a group crossed product by $\Gamma$ [1309.2363].

These corner realizations do not identify an explicit invertible intertwiner in the Hilbert-space sense. A plausible implication is that they provide a categorical analogue of similarity: semigroup crossed products defined by partial isometries can be analyzed inside group crossed products, and many representation-theoretic and ideal-theoretic invariants may be transferred along the resulting Morita equivalence [1710.06708].

## 6. Inverse-semigroup and geometric analogues

Partial-isometric models also arise at the level of semigroup $C^*$-algebras and inverse semigroups. For a cancellative semigroup $P$, the 2020 construction defines a universal $C^*$-algebra from partial isometric representations, generalizing isometric models. When $P$ is an LCM monoid, the resulting algebra satisfies
$$
C^*(P)\cong C^*(S_P)\cong C^*(\mathcal{G}_u(S_P)),
$$
where $S_P$ is an inverse semigroup canonically attached to $P$ [2001.00156]. The partial-isometric construction treats left and right ideal structure symmetrically and contains the isometric theory as a special case.

Inverse semigroups of concrete partial isometries furnish explicit algebraic shadows of group similarity. The semigroup $\mathbf{I}\mathbb{N}_\infty$ of partial co-finite isometries of the positive integers is simple, $E$-unitary, and $F$-inverse; its least group quotient is isomorphic to $(\mathbb{Z},+)$ [1904.06638]. The semigroup $\mathbf{ID}_\infty$ of partial isometries of $\mathbb{Z}$ with cofinite domain and image is also $F$-inverse, the quotient by the minimum group congruence is isomorphic to the full isometry group $\mathrm{Iso}(\mathbb{Z})$, and
$$
\mathbf{ID}_\infty \cong \mathrm{Iso}(\mathbb{Z})\ltimes_{\mathfrak h}\mathscr{P}_{\!\infty}(\mathbb{Z})
$$
[1904.06644]. In both cases, the least group congruence extracts a maximal group image from a semigroup of partial symmetries.

There are also geometric analogues in which “similarity” is encoded by partial bijections of boundary objects rather than by operator conjugation. For metrics on doubles of a metric space, the coarse-equivalence classes of compatible metrics form an inverse semigroup, and inverse subsemigroups attached to families of isometric subspaces project onto semigroups of partial bijections of the associated visual boundary [2212.01863]. In Euclidean space this yields a split semigroup surjection onto partial bijections induced by restrictions of orthogonal operators, while for rooted trees it yields a split semigroup surjection onto partial bijective locally bi-Lipschitz maps of the boundary [2212.01863]. In a different geometric direction, finitely generated semigroups of real Möbius transformations are semidiscrete and inverse free exactly when every composition sequence converges ideally to the boundary; the same work gives a complete two-generator classification and analyzes the relation between the “group part” of a semigroup and equality of forward and backward limit sets [1609.00576]. These constructions suggest that, beyond Hilbert-space similarity, the subject also includes group-completion, inverse-semigroup, and boundary-dynamical versions of the same organizing principle.

A general pattern emerges from these results. True similarity to isometric semigroups is governed by two-sided norm control and compatible projection structure; dilation theory isolates the obstruction in explicitly describable components; and in $C^*$-algebraic or inverse-semigroup settings, the closest analogue is often a full-corner realization, a least group quotient, or a groupoid model. The unifying theme is that semigroup behavior becomes tractable precisely when it can be embedded, dilated, or compared to an isometric or group-like framework, while the failures of total order, inverse structure, commutative idempotents, or lower norm bounds produce the main obstructions.

Source: https://www.emergentmind.com/topics/similarity-to-isometric-semigroups