Similarity to Isometric Semigroups

• Similarity to isometric semigroups is the process of transforming semigroup representations into models acting by isometries through appropriate norm bounds and coordinate changes.
• Operator-theoretic methods, including two-sided norm bounds and weighted averaging techniques, establish isometric similarity in both continuous and discrete settings.
• Dilation theory and full-corner constructions in C*-algebra frameworks bridge semigroup representations with unitary, inverse, and Morita equivalent models.

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 (Popov, 2013), similarity of Hilbert-space $C_0$-semigroups to isometric semigroups via uniform two-sided norm bounds (Oliva-Maza et al., 31 Aug 2025), Følner-type criteria for amenable semigroups (Badea et al., 2018), dilation and Wold-type decompositions for semigroup representations (Li, 2017, Li, 2021), 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 (Zahmatkesh, 2017). 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 (Oliva-Maza et al., 31 Aug 2025). 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 $C^*$0 and a densely defined operator $C^*$1 such that, for some $C^*$2,

$C^*$3

or the same with $C^*$4 (Oliva-Maza et al., 31 Aug 2025). A resolvent counterpart replaces time averages by Abel-type integrals of $C^*$5, 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 $C^*$6 is similar to an isometry if there exist $C^*$7 such that

$C^*$8

for every $C^*$9 and every $C_0$0 (Badea et al., 2018). 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 $C_0$1, imply similarity to an isometric representation (Badea et al., 2018). 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 $C_0$2 is an irreducible norm closed semigroup of complex $C_0$3 matrices, then the following are equivalent: $C_0$4 is simultaneously similar to a semigroup of partial isometries; all idempotents in $C_0$5 commute and every spectrum satisfies $C_0$6; and all idempotents in $C_0$7 commute while there exist constants $C_0$8 such that

$C_0$9

for every nonzero $C_0$0 (Popov, 2013). 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 $C_0$1 and norm $C_0$2 on nonzero elements whose norm closure contains non-commuting idempotents, so the semigroup is not similar to a semigroup of partial isometries (Popov, 2013). 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 (Popov et al., 2013). In the irreducible case, every such semigroup acts as a family of generalized weighted composition operators on a space $C_0$3. 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 (Popov et al., 2013). 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 $C_0$4, the following are equivalent: $C_0$5 is a positive cone; all $C_0$6-algebras generated by faithful isometric non-unitary representations of $C_0$7 are canonically isomorphic; all such representations are inverse; and, for every such representation $C_0$8 and all relevant semigroup elements,

$C_0$9

(Aukhadiev et al., 2012). The same source states the corollary that the universal semigroup $\mathcal{T}=(T(t))_{t\ge 0}$0-algebra coincides with the reduced semigroup $\mathcal{T}=(T(t))_{t\ge 0}$1-algebra if and only if $\mathcal{T}=(T(t))_{t\ge 0}$2 is totally ordered.

The case $\mathcal{T}=(T(t))_{t\ge 0}$3 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 (Aukhadiev et al., 2012). 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 $\mathcal{T}=(T(t))_{t\ge 0}$4 provides a complementary example. Its non-unitary irreducible isometric inverse representations are classified up to unitary equivalence by two models, $\mathcal{T}=(T(t))_{t\ge 0}$5 and $\mathcal{T}=(T(t))_{t\ge 0}$6, and every isometric inverse representation decomposes as

$\mathcal{T}=(T(t))_{t\ge 0}$7

where $\mathcal{T}=(T(t))_{t\ge 0}$8 is unitary (Grigoryan et al., 2013). At the same time, there is a continuum of non-inverse irreducible isometric $\mathcal{T}=(T(t))_{t\ge 0}$9-representations, with $\alpha,\beta>0$0 inverse if and only if $\alpha,\beta>0$1 or $\alpha,\beta>0$2 (Grigoryan et al., 2013). 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 $\alpha,\beta>0$3 of a right LCM semigroup, three conditions are equivalent: $\alpha,\beta>0$4 has a $\alpha,\beta>0$5-regular dilation; $\alpha,\beta>0$6 has a minimal isometric Nica-covariant dilation; and, for every finite subset $\alpha,\beta>0$7,

$\alpha,\beta>0$8

This Brehmer-type condition gives an explicit positivity test for passage to an isometric Nica-covariant model (Li, 2017). 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 $\alpha,\beta>0$9 of the odometer semigroup, there is a unique decomposition

$$\alpha \|h\| \leq \|T(t)h\| \leq \beta \|h\|, \qquad \forall h\in H,\ \forall t\ge 0.$$0

where the four summands encode unitary-row unitary, unitary-pure, pure-row unitary, and weak bi-shift behavior (Li, 2021). 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

$$\alpha \|h\| \leq \|T(t)h\| \leq \beta \|h\|, \qquad \forall h\in H,\ \forall t\ge 0.$$1

(Li, 2021). 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 $$\alpha \|h\| \leq \|T(t)h\| \leq \beta \|h\|, \qquad \forall h\in H,\ \forall t\ge 0.$$2-isometric $$\alpha \|h\| \leq \|T(t)h\| \leq \beta \|h\|, \qquad \forall h\in H,\ \forall t\ge 0.$$3-semigroups, the generator is characterized by a Lumer–Phillips type theorem involving closedness, surjectivity, quasidissipativity, and $$\alpha \|h\| \leq \|T(t)h\| \leq \beta \|h\|, \qquad \forall h\in H,\ \forall t\ge 0.$$4-skew-symmetry (Rydhe, 2018). However, for $$\alpha \|h\| \leq \|T(t)h\| \leq \beta \|h\|, \qquad \forall h\in H,\ \forall t\ge 0.$$5, these semigroups are not generally similar to isometric ones. The prototypical analytic $$\alpha \|h\| \leq \|T(t)h\| \leq \beta \|h\|, \qquad \forall h\in H,\ \forall t\ge 0.$$6-isometry $$\alpha \|h\| \leq \|T(t)h\| \leq \beta \|h\|, \qquad \forall h\in H,\ \forall t\ge 0.$$7 on the Dirichlet space is not similar to any isometry on a Hilbert space because its spectrum is the closed unit disk (Rydhe, 2018). 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 $$\alpha \|h\| \leq \|T(t)h\| \leq \beta \|h\|, \qquad \forall h\in H,\ \forall t\ge 0.$$8-algebraic setting, similarity to isometric semigroups is often replaced by a corner or Morita-equivalence construction. Let $$\alpha \|h\| \leq \|T(t)h\| \leq \beta \|h\|, \qquad \forall h\in H,\ \forall t\ge 0.$$9 be the positive cone of a totally ordered abelian discrete group $\mathcal{T}$0, and let $\mathcal{T}$1 be an action of $\mathcal{T}$2 by extendible endomorphisms of a $\mathcal{T}$3-algebra $\mathcal{T}$4. The main theorem of the 2017 crossed-product paper states that the partial-isometric crossed product satisfies

$\mathcal{T}$5

where $\mathcal{T}$6 is a subalgebra of $\mathcal{T}$7 generated by faithful copies of $\mathcal{T}$8, $\mathcal{T}$9 is induced by shift on $E$0, and $E$1 is a projection in the multiplier algebra (Zahmatkesh, 2017). Consequently, the partial-isometric crossed product is Morita equivalent to a group crossed product. The same paper identifies an essential ideal $E$2 in $E$3, namely the kernel of the surjection onto the isometric crossed product, and shows that $E$4 is a full corner in an ideal $E$5 of $E$6 (Zahmatkesh, 2017).

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 $E$7, and there is always a natural surjection

$E$8

Its kernel is explicitly described, and for $E$9 or $C_0$0 it is a full corner of the compact operators on the Hilbert module (Adji et al., 2013). 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 $C_0$1 (Adji et al., 2013).

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 (Zahmatkesh, 2017).

6. Inverse-semigroup and geometric analogues

Partial-isometric models also arise at the level of semigroup $C_0$2-algebras and inverse semigroups. For a cancellative semigroup $C_0$3, the 2020 construction defines a universal $C_0$4-algebra from partial isometric representations, generalizing isometric models. When $C_0$5 is an LCM monoid, the resulting algebra satisfies

$C_0$6

where $C_0$7 is an inverse semigroup canonically attached to $C_0$8 (Starling et al., 2020). 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 $C_0$9 of partial co-finite isometries of the positive integers is simple, $C^*$00-unitary, and $C^*$01-inverse; its least group quotient is isomorphic to $C^*$02 (Gutik et al., 2019). The semigroup $C^*$03 of partial isometries of $C^*$04 with cofinite domain and image is also $C^*$05-inverse, the quotient by the minimum group congruence is isomorphic to the full isometry group $C^*$06, and

$C^*$07

(Gutik et al., 2019). 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 (Manuilov, 2022). 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 (Manuilov, 2022). 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 (Jacques et al., 2016). 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^*$08-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.