Full Left Hilbert Algebras in Operator Theory

• Full left Hilbert algebras are involutive algebras with a Hilbert-space structure that internalize bounded elements, ensuring A equals its bounded completion.
• They organize both bounded and unbounded multipliers via left and right regular representations, linking these to the associated von Neumann algebras.
• Their role in free-Poisson Fock functors translates algebraic input into modular field operators and free cumulant structures in noncommutative probability.

Full left Hilbert algebras belong to the operator-algebraic branch of Hilbert algebra theory, not to the implication-algebra usage common in algebraic logic. In the formulations represented here, they are involutive algebras equipped with a Hilbert-space structure and a compatible left regular representation, and “fullness” expresses that the natural bounded or left-bounded completion data are already internal to the algebra. In Dixmier-style language, a Hilbert algebra is full when it coincides with its fulfillment $\mathbf A_{\mathrm b}$ of bounded elements, while in the pseudo left Hilbert algebra framework one says that $A$ is full when $A=A''$, where $A''=D(S)\cap\mathfrak B$ (Goursac, 2014, Yang, 3 Apr 2025).

1. Definition and operator-theoretic framework

A left Hilbert algebra, in the formulation explicitly recalled in the free-Poisson framework, is an associative involution algebra $A$ over $\mathbb C$ with involution $S:A\to A$ and inner product $\langle\cdot,\cdot\rangle$ such that, for each $\xi\in A$, the left multiplication

$\pi_l(\xi):A\ni \eta\mapsto \xi\eta$

is bounded, the compatibility condition

$A$0

holds for all $A$1, the involution $A$2 is preclosed, and

$A$3

is dense in $A$4 (Yang, 3 Apr 2025). This is the setting in which the Tomita–Takesaki data of the algebraic core are built into the one-particle Hilbert space.

A closely related formulation, used in multiplier theory, starts from a complex $A$5-algebra $A$6 with scalar product $A$7 satisfying

$A$8

continuity of the maps $A$9, and density of the set $A=A''$0 in $A=A''$1 (Goursac, 2014). The completion for the Hilbert norm is denoted $A=A''$2, and the scalar product convention is left antilinear and right linear.

In both languages, the left and right regular representations are fundamental. For $A=A''$3 one has

$A=A''$4

and both extend to bounded operators on $A=A''$5. Their weak closures define the left and right von Neumann algebras of the Hilbert algebra, with

$A=A''$6

(Goursac, 2014). In the standard left-Hilbert-algebra setting attached to a normal semifinite faithful weight $A=A''$7 on a von Neumann algebra $A=A''$8, the canonical example is

$A=A''$9

with multiplication

$A''=D(S)\cap\mathfrak B$0

and involution

$A''=D(S)\cap\mathfrak B$1

(Yang, 3 Apr 2025).

2. Fulfillment, bounded vectors, and fullness

The operator-algebraic literature represented here uses two closely related fullness constructions. In the Dixmier Hilbert algebra setting, an element $A''=D(S)\cap\mathfrak B$2 is called bounded if there exists $A''=D(S)\cap\mathfrak B$3 such that

$A''=D(S)\cap\mathfrak B$4

or equivalently there exists $A''=D(S)\cap\mathfrak B$5 such that

$A''=D(S)\cap\mathfrak B$6

The set of bounded elements is denoted $A''=D(S)\cap\mathfrak B$7 and is called the fulfillment of $A''=D(S)\cap\mathfrak B$8; the algebra is full if

$A''=D(S)\cap\mathfrak B$9

Moreover, $A$0 is again a Hilbert algebra with the same Hilbert completion (Goursac, 2014).

In the pseudo left Hilbert algebra language, one lets $A$1 be the set of left bounded vectors in $A$2, $A$3 the set of right bounded vectors, and defines

$A$4

A pseudo left Hilbert algebra is then called full if

$A$5

This formulation is explicitly used in the free-Poisson construction, where one of the structural statements is that

$A$6

so the resulting von Neumann algebra is insensitive to replacing $A$7 by its full closure (Yang, 3 Apr 2025).

These definitions isolate the same operator-theoretic phenomenon: the algebraic core may be strictly smaller than the natural domain determined by bounded left multiplication and the closed involution. A common misconception is that fullness is merely a technical regularity condition. The cited work shows instead that fullness controls whether the bounded part of the regular representation has already been internalized by the algebra itself (Goursac, 2014). A plausible implication is that full left Hilbert algebras are the natural domain on which left multiplication, modular structure, and subsequent functorial constructions can be treated without repeatedly passing to closures.

3. Multipliers, commutants, and unbounded operator structure

The paper on multipliers develops the bounded and unbounded operator theory naturally attached to a Hilbert algebra. A bounded multiplier of $A$8 is a pair

$A$9

such that

$\mathbb C$0

The set of bounded multipliers is denoted $\mathbb C$1 (Goursac, 2014). The key identification is

$\mathbb C$2

and consequently

$\mathbb C$3

Thus bounded multipliers recover the same semifinite von Neumann algebra as the left regular representation.

The one-sided viewpoint is equally explicit. For bounded operators $\mathbb C$4, the following are equivalent:

1. $\mathbb C$5,
2. $\mathbb C$6 is a bounded left multiplier and $\mathbb C$7,
3. $\mathbb C$8 is a bounded right multiplier and $\mathbb C$9, where $S:A\to A$0 is the involution extended to the Hilbert completion (Goursac, 2014). This expresses the standard principle that the right action is canonically determined by the left action and the involution.

The unbounded theory is substantially richer. An unbounded multiplier is a pair

$S:A\to A$1

satisfying

$S:A\to A$2

The set $S:A\to A$3 is an $S:A\to A$4-algebra on the common domain $S:A\to A$5, and the embedding

$S:A\to A$6

makes $S:A\to A$7 a $S:A\to A$8-ideal in $S:A\to A$9 (Goursac, 2014). The closures of unbounded multipliers are not arbitrary closable operators: for every $\langle\cdot,\cdot\rangle$0, the closure

$\langle\cdot,\cdot\rangle$1

is affiliated with the von Neumann algebra $\langle\cdot,\cdot\rangle$2. In addition,

$\langle\cdot,\cdot\rangle$3

and $\langle\cdot,\cdot\rangle$4 is a pre-GW$\langle\cdot,\cdot\rangle$5-algebra (Goursac, 2014).

For full left Hilbert algebras, these results are decisive because they show that fullness does not terminate at the bounded envelope. It also organizes the natural unbounded operators into a bicommutant-stable algebra affiliated with the left von Neumann algebra. This is the operator-algebraic sense in which full left Hilbert algebras generate both bounded and unbounded completion theories.

4. Full left Hilbert algebras as input to free-Poisson Fock functors

The free-Poisson construction places full left Hilbert algebras at the center of a Fock-space functor. For a Hilbert space $\langle\cdot,\cdot\rangle$6, the full Fock space is

$\langle\cdot,\cdot\rangle$7

with vacuum vector $\langle\cdot,\cdot\rangle$8 (Yang, 3 Apr 2025). For $\langle\cdot,\cdot\rangle$9, the left creation operator and its adjoint are

$\xi\in A$0

$\xi\in A$1

For a densely defined operator $\xi\in A$2 on $\xi\in A$3, the preservation operator is

$\xi\in A$4

If $\xi\in A$5 is a pseudo left Hilbert algebra and $\xi\in A$6, the three basic components are

$\xi\in A$7

and the fundamental field operator is

$\xi\in A$8

It satisfies

$\xi\in A$9

The von Neumann algebra generated by the free-Poisson fields is

$\pi_l(\xi):A\ni \eta\mapsto \xi\eta$0

(Yang, 3 Apr 2025).

When $\pi_l(\xi):A\ni \eta\mapsto \xi\eta$1 is the left Hilbert algebra attached to a pair $\pi_l(\xi):A\ni \eta\mapsto \xi\eta$2, one writes

$\pi_l(\xi):A\ni \eta\mapsto \xi\eta$3

and the map

$\pi_l(\xi):A\ni \eta\mapsto \xi\eta$4

is the centered free Poisson random weight. The uncentered version on $\pi_l(\xi):A\ni \eta\mapsto \xi\eta$5 is

$\pi_l(\xi):A\ni \eta\mapsto \xi\eta$6

The construction is explicitly described as “a free Poisson type functor for left Hilbert algebras similar to Voiculescu's free Gaussian functor for Hilbert spaces” (Yang, 3 Apr 2025).

The Poisson character of the functor is encoded by its free cumulants. For $\pi_l(\xi):A\ni \eta\mapsto \xi\eta$7,

$\pi_l(\xi):A\ni \eta\mapsto \xi\eta$8

while for the uncentered fields,

$\pi_l(\xi):A\ni \eta\mapsto \xi\eta$9

(Yang, 3 Apr 2025). Thus multiplication moments in the input left Hilbert algebra become free cumulants in the Fock-space output.

The output algebra retains a precise modular structure. The vacuum vector $A$00 is cyclic and separating for $A$01, and on tensor words one has

$A$02

$A$03

At the von Neumann algebraic level, the paper proves the factor criterion

$A$04

and, when $A$05, gives explicit free-product decompositions (Yang, 3 Apr 2025).

5. Pseudo left Hilbert algebras, degeneracy, and decomposition theory

The free-Poisson framework first enlarges the input category from left Hilbert algebras to pseudo left Hilbert algebras. A pseudo left Hilbert algebra satisfies the boundedness of $A$06, the compatibility relation

$A$07

and preclosedness of $A$08, but drops the density condition on $A$09 (Yang, 3 Apr 2025). This “degenerate” variant allows nontrivial vectors with trivial multiplication.

Its basic structure theorem is the orthogonal decomposition

$A$10

with multiplication

$A$11

for $A$12 and $A$13 (Yang, 3 Apr 2025). The first summand is the genuine left-Hilbert-algebra part; the orthogonal complement carries trivial multiplication.

Under the $A$14-construction this yields a free-product decomposition,

$A$15

and the trivial-multiplication sector becomes the free Araki–Woods part of the theory (Yang, 3 Apr 2025). In this sense, free Araki–Woods algebras appear as special cases of free Poisson algebras for degenerate left Hilbert algebras.

The same decomposition principle is used probabilistically. The paper states that the Lévy–Itô decomposition of a jointly freely infinitely divisible family can be interpreted as a decomposition of a degenerate left Hilbert algebra, and that any additive time-parameterized free Lévy process admits a realization as unbounded operators in a full Fock space. As an application, the filtration algebras of any additive free Lévy process are always interpolated group factors with a possible additional atom (Yang, 3 Apr 2025).

A common misunderstanding is that the degenerate extension weakens the theory only algebraically. The decomposition theorem shows something sharper: degeneracy isolates precisely the sector in which multiplication disappears and only the modular one-particle structure remains. That is why the free-Poisson and free-Araki–Woods constructions can be treated within one framework.

6. Terminological scope and distinction from algebraic-logical Hilbert algebras

The expression “Hilbert algebra” is heavily overloaded. In several adjacent arXiv literatures it denotes implication algebras of type $A$16, such as algebras

$A$17

satisfying implicational axioms, prelinear Hilbert algebras, frontal Hilbert algebras, or skew Hilbert algebras (Jansana et al., 2018, Castiglioni et al., 2018, Chajda et al., 2021). Those papers develop categorical adjunctions to implicative semilattices or Gödel algebras, or generalize implication to non-distributive logics, but they do not define left Hilbert algebras or full left Hilbert algebras in the operator-algebraic sense (Jansana et al., 2018, Castiglioni et al., 2018, Chajda et al., 2021).

This distinction matters because fullness has a different semantic role in the two traditions. In the operator-algebraic setting, fullness concerns bounded vectors, regular representations, and closure under the natural involutive Hilbert-space structure (Goursac, 2014, Yang, 3 Apr 2025). In the algebraic-logical setting, the central problems concern implication, filters, adjunctions to Heyting or implicative-semilattice categories, or structural generalizations such as skew Hilbert algebras (Jansana et al., 2018, Castiglioni et al., 2018, Chajda et al., 2021).

For research use, “full left Hilbert algebra” should therefore be read as an operator-algebraic notion tied to Tomita–Takesaki theory, regular representations, multiplier algebras, and Fock-space constructions. Within that setting, the main structural facts established here are that fullness can be encoded as

$A$18

that bounded multipliers recover the left von Neumann algebra,

$A$19

that unbounded multipliers form a pre-GW$A$20-algebra affiliated with this bounded envelope, and that full left Hilbert algebras provide the natural input datum for the free-Poisson functor

$A$21

(Goursac, 2014, Yang, 3 Apr 2025).